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Acoplamentos: uma primeira visão e algumas aplicações
(Universidade Federal do Espírito Santo, 2016-06-24) Oliveira, Weverthon Lobo de; Valentim, Fábio Júlio da Silva; Gonçalves Junior, Etereldes; Romero, Freddy Rolando Hernandez
We will cover the Couplings Theory showing applications in Probability and analysis of problems, more specifically, exhibit applications in two contexts, which are the Markov Chain and Optmal Transport Problem. Chapter 1 presents some preliminary ideas which serve as a base for understanding this work, we will cover Probability notions, Markov Chains, Topology, Continuous Functions and semicontinuous and Convex Analysis. In the next two chapters will be presented the main contexts with their respective applications, more precisely, reserve the Chapter 2 to display Couplings and some of its applications in Markov chains, highlighting the calculation of mixing time of a Markov Chain and the proof of Theorem of Convergence via couplings. Chapter 3 will discuss the Transportation Problem Great, which can be divide in two problems, the Monge problem and Kantorovich problem. the difference will be presented between the two problems, conditions for solution of existence and the relationship between the two problems and finally present an suggestion algorithm that solves Kantorovich problem and demonstrate the isoperimetric inequality via Monge problem.